Solve x+6/2x+10+5/x^2-25=1y

$$\frac { x + 6 } { 2 x + 10 } + \frac { 5 } { x ^ { 2 } - 25 } = 1 y$$

Solve for y (complex solution)

$$y=-\frac{x-4}{2\left(5-x\right)}$$

$$x\neq -5\text{ and }x\neq 5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(x-5\right)\left(x+5\right), the least common multiple of 2x+10,x^{2}-25.

$$\left(x-5\right)\left(x+6\right)+2\times 5=1y\times 2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by x+6 and combine like terms.

$$x^{2}+x-30+2\times 5=1y\times 2\left(x-5\right)\left(x+5\right)$$

Multiply 2 and 5 to get 10.

$$x^{2}+x-30+10=1y\times 2\left(x-5\right)\left(x+5\right)$$

Add -30 and 10 to get -20.

$$x^{2}+x-20=1y\times 2\left(x-5\right)\left(x+5\right)$$

Multiply 1 and 2 to get 2.

$$x^{2}+x-20=2y\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply 2y by x-5.

$$x^{2}+x-20=\left(2yx-10y\right)\left(x+5\right)$$

Use the distributive property to multiply 2yx-10y by x+5 and combine like terms.

$$x^{2}+x-20=2yx^{2}-50y$$

Swap sides so that all variable terms are on the left hand side.

$$2yx^{2}-50y=x^{2}+x-20$$

Combine all terms containing y.

$$\left(2x^{2}-50\right)y=x^{2}+x-20$$

Divide both sides by -50+2x^{2}.

$$\frac{\left(2x^{2}-50\right)y}{2x^{2}-50}=\frac{\left(x-4\right)\left(x+5\right)}{2x^{2}-50}$$

Dividing by -50+2x^{2} undoes the multiplication by -50+2x^{2}.

$$y=\frac{\left(x-4\right)\left(x+5\right)}{2x^{2}-50}$$

Divide \left(-4+x\right)\left(5+x\right) by -50+2x^{2}.

$$y=\frac{x-4}{2\left(x-5\right)}$$

Solve for y

$$y=-\frac{x-4}{2\left(5-x\right)}$$

$$|x|\neq 5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(x-5\right)\left(x+5\right), the least common multiple of 2x+10,x^{2}-25.

$$\left(x-5\right)\left(x+6\right)+2\times 5=1y\times 2\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply x-5 by x+6 and combine like terms.

$$x^{2}+x-30+2\times 5=1y\times 2\left(x-5\right)\left(x+5\right)$$

Multiply 2 and 5 to get 10.

$$x^{2}+x-30+10=1y\times 2\left(x-5\right)\left(x+5\right)$$

Add -30 and 10 to get -20.

$$x^{2}+x-20=1y\times 2\left(x-5\right)\left(x+5\right)$$

Multiply 1 and 2 to get 2.

$$x^{2}+x-20=2y\left(x-5\right)\left(x+5\right)$$

Use the distributive property to multiply 2y by x-5.

$$x^{2}+x-20=\left(2yx-10y\right)\left(x+5\right)$$

Use the distributive property to multiply 2yx-10y by x+5 and combine like terms.

$$x^{2}+x-20=2yx^{2}-50y$$

Swap sides so that all variable terms are on the left hand side.

$$2yx^{2}-50y=x^{2}+x-20$$

Combine all terms containing y.

$$\left(2x^{2}-50\right)y=x^{2}+x-20$$

Divide both sides by -50+2x^{2}.

$$\frac{\left(2x^{2}-50\right)y}{2x^{2}-50}=\frac{\left(x-4\right)\left(x+5\right)}{2x^{2}-50}$$

Dividing by -50+2x^{2} undoes the multiplication by -50+2x^{2}.

$$y=\frac{\left(x-4\right)\left(x+5\right)}{2x^{2}-50}$$

Divide \left(-4+x\right)\left(5+x\right) by -50+2x^{2}.

$$y=\frac{x-4}{2\left(x-5\right)}$$

Solve for x (complex solution)

$$x=-\frac{2\left(5y-2\right)}{1-2y}$$

$$y\neq \frac{9}{20}\text{ and }y\neq \frac{1}{2}$$

Solve for x

$$x=-\frac{2\left(5y-2\right)}{1-2y}$$

$$y\neq \frac{1}{2}\text{ and }y\neq \frac{9}{20}$$